Carraher et al.'s odd-even density conjecture for three-element distance sets

Let $D=\{1,j,k\}$, where $j$ is odd, $j\geqslant 3$, and $k$ is even. Here $\mu(D)$ denotes the supremum of the densities of sequences of non-negative integers whose pairwise differences do not lie in $D$.

Carraher et al.'s odd-even density conjecture. If $k\geqslant 3j$, then

$$\mu(D)=\frac{k}{2(k+j)}.$$

This conjecture was posed as part of a program determining the density of sequences avoiding prescribed separations. The supplied text does not state whether it has been proved or disproved.